Optimal reinsurance under risk and uncertainty

Abstract

This paper deals with the optimal reinsurance problem if both insurer and reinsurer are facing risk and uncertainty, though the classical uncertainty free case is also included. The insurer and reinsurer degrees of uncertainty do not have to be identical. The decision variable is not the retained (or ceded) risk, but its sensitivity with respect to the total claims. Thus, if one imposes strictly positive lower bounds for this variable, the reinsurer moral hazard is totally eliminated. Three main contributions seem to be reached. Firstly, necessary and sufficient opti- mality conditions are given. Secondly, the optimal contract is often a bang-bang solution, i:e:, the sensitivity between the retained risk and the total claims saturates the imposed constraints. For some special cases the optimal contract might not be bang-bang, but there is always a bang-bang contract as close as desired to the optimal one. Thirdly, the optimal reinsurance problem is equivalent to other linear programming problem, despite the fact that risk, uncertainty, and many premium principles are not linear. This may be impor- tant because linear problems are easy to solve in practice, since there are very efficient algorithms.